Profinite Separation Systems

The author briefly reviews some general aspects of Bose condensation and off-diagonal long range order in infinite bulk systems and then discusses to what extent these features might be manifested in finite systems with free surfaces such as droplets of liquid [sup 4]He at low temperature. The author focuses on the structure of the one-body density matrix and the macroscopic wave function in the surface region, and notes the possibility of a surface condensate.

The Kolmogorov-Avrami-type formulation for both the latent nucleation (inhomogeneous) and the random nucleation (homogeneous) cases is presented for finite systems. In finite systems, there appear three time regimes. For the latent nucleus case the fraction of transformed regions remains as 1- e - N V ( N : density of nuclei, V : system volume) and for the random nucleation case it approaches unity in the time dependence as 1- A e - J V t ( A : constant, J : nucleation rate). The growth rate in finite systems is always smaller than that in infinite systems. We present the numerical results of one-dimensional system as an example.

We consider finite macroscopic systems, i.e. systems of large but finite degrees of freedom, which we believe are poorly understood as compared with small systems and infinite systems. We focus on pure states that do not have the "cluster property". Such a pure state is entangled macroscopically, and is quite anomalous in view of many-body physics because it does not approach any pure state in the infinite-size limit. However, we often encounter such anomalous states when studying finite macroscopic systems, such as quantum computers with many qubits and finite systems that will exhibit symmetry breaking in the infinite-size limit. We study stabilities of such anomalous states in general systems. In contrast to the previous works, we obtain general and universal results, by making full use of the locality of the theory. Using the general results, we discuss roles of anomalous states in quantum computers, and the mechanism of emergence of a classical world from quantum theory.

We study the problem of relating the long time behavior of finite and infinite systems of locally interacting components. We consider in detail a class of lincarly interacting diffusionsx(t)={x i (t),i ∈ ℤ d } in the regime where there is a one-parameter family of nontrivial invariant measures. For these systems there are naturally defined corresponding finite systems, $$x^N (t) = \left\{ {x_i^N (t),i \in \Lambda _N } \right\}$$ , with $$\Lambda _N = ( - N,N]^d \cap \mathbb{Z}^d$$ . Our main result gives a comparison between the laws ofx(t N ) andx N (t N ) for timest N →∞ asN→∞. The comparison involves certain mixtures of the invariant measures for the infinite system.

We present infinite analogues of our splinter lemma for constructing nested sets of separations. From these we derive several tree-of-tangles-type theorems for infinite graphs and infinite abstract separation systems.

The singularities of the pressure as a function of the activity z are examined by a certain limiting procedure applicable to the solved hard-hexagon model (Baxter 1980) and the tempered hard-square model (Fisher 1963). For hard hexagons the only singularities found in the finite complex z plane are two isolated branch points on the real z axis. For the tempered hard squares, the only singularities found are three isolated branch points on the real z axis. By the same procedure applied to the Ising model, the singularities of the energy as a function of temperature are found to be isolated branch points in the complex temperature plane. Comparison with numerical results shows that the dense distribution of singularities of finite systems disappears in the finite system limit, save for a few isolated points.

Unwinding relations have been widely used to prove that finite systems are secure with respect to a variety of noninterference policies. The latter are prominent instances of security-relevant hyperproperties. As hyperproperties are defined on potentially infinite systems, a new mathematical development is needed in order to (re)use unwinding relations for generic verification of security-relevant hyperproperties. In this paper we propose a framework for coinductive unwinding of security relevant hyperproperties. To illustrate the usefulness of the framework, we show that Mantel’s Basic Security Predicates (BSPs), the noninterference policies they compose, as well as their respective unwinding relations, have a meaningful coinductive reinterpretation. We prove that in a number of cases the coinductive variants of the unwinding relations imply the respective coinductive variants of the BSPs. Moreover, the latter can be used to compose high-level security-relevant hyperproperties for both finite and infinite systems. A number of the unwinding theorems also hold as expected. In conclusion, the proposed framework and results are useful both theoretically in the study of hyperproperties and in practice for verification of hyperproperties on potentially infinite systems.

This paper was prepared for the 41st Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Dallas, Tex., Oct. 2–5, 1966. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract The purpose of this paper is to show that data from a drill-stem test may be effectively used to aid in the design of production stimulation treatments. Formation characteristics determined from the pressure behavior on a tests are compared to simulated characteristics by propping or etching a core sample to indicate the type and size of treatment necessary. The formation characteristics considered that may be used in stimulation design are production rate, permeability, flow capacity, and reservoir pressure and its stability. Formation conditions are investigated to show how they may affect the stimulation treatment. These conditions are existing fracture system, permeability discontinuities, and formation damage. These formation conditions and characteristics are used to show an estimated production increase wider certain conditions. Field results are used to illustrate the method. Introduction Well stimulation is the source of a good part of the oil industry's domestic production. There is a constant search for methods and supporting formation characteristics to help design stimulation treatments to give optimum results. To get the best possible results, a stimulation treatment should be designed utilizing all pertinent data available. The most reliable single piece of information available is a pressure analysis of the drill-stem test. This is the first evidence of the performance of the well. The pressure performance of a drill-stem test may provide natural or matrix production, flow capacity, permeability, formation damage, formation anomalies, and reservoir pressure and its stability. These data may be evaluated and compared to an improved or stimulated condition to determine the need for and the type of treatment necessary, and the production increase that may be expected, In addition, this comparison may indicate the size and rate of treatment to accomplish a certain condition. However, the design of the stimulation treatment is beyond the scope of this paper. BACKGROUND DEVELOPMENT There are two primary classifications of a reservoir in pressure analysis those in an infinite system and those in a finite system. An infinite system is one in which the outer boundary is an infinite distance from the wellbore and has not been reached on the test. Therefore, constant pressure is maintained at the extremities of the pressure disturbance. This condition exists in most cases during a test; therefore, only the infinite system is considered in this paper. Frequently, a boundary is reached; however, the pressure behavior before this disturbance may be used to determine formation characteristics.

This article studies the strong conical hull intersection property (strong CHIP) for a general (possibly infinite) system of convex sets in normed linear spaces. The first part of the article presents necessary and sufficient conditions for strong CHIP in terms of tangent cones and feasible direction cones in the primal space, and reveals some differences between strong CHIP for finite and infinite systems. The second part of the paper proves that strong CHIP has the segment extension property, i.e. if strong CHIP is satisfied at some ‘base points’ then it can be extended from the base points along line segments to the entire set. Thus, the verification effort of strong CHIP can be greatly reduced in many cases. For example, if the intersection set is a translated cone, then strong CHIP at the vertex ensures strong CHIP for the entire cone. In the special case of Euclidean space, strong CHIP at the extreme points of the intersection set ensures strong CHIP for the entire intersection.

A fundamental problem of x-ray spectroscopy is the role of relaxation of the electronic subsystem in the field of the transient core hole. The main intention of the present study is to explore the dynamics due to core-hole relaxation in the whole time domain, and to find out how it is manifested in finite molecular systems in comparison with solids. A technique is developed based on a reduction of the Nozi\'eres-De Dominicis equation to a set of linear algebraic equations. The developed time-dependent formalism is applied to a numerical investigation of a one-dimensional tight-binding model. The formation of the x-ray profiles is explored on the real time scale, and the role of interaction with the core hole, band filling, and the final-state rule are investigated for systems of different size. The formation of spectra of the infinite translational invariant system is studied by extensions of the finite systems. We found that the dynamics of finite systems, like molecules, differs qualitatively from solids: Contrary to the latter the time lapse of the Nozi\'eres-De Dominicis domain for finite systems is squeezed between the inverse bandwidth and the revival time, which is proportional to the system size. For small molecules this means that there is no time for a ``Mahan-Nozi\'eres-De Dominicis singularity'' to develop. Comparison with the strict solution of the Nozi\'eres-De Dominicis equation shows that the adiabatic approximation describes x-ray absorption and emission considerably better than the fast approximation. This explains the suppression of the relaxation effects in x-ray emission of, e.g., gas phase and surface adsorbed molecules, but also that these effects are essential for the absorption case. There is still a quantitative distinction between the adiabatic approximation and the strict approach, which becomes more important for larger systems. Adopting the so-called finite state rule by von Barth and Grossman also for molecules, an almost complete numerical agreement between this rule and the strict x-ray-absorption and emission profiles for systems of different sizes is obtained. The simulations indicate that the final-state rule correction is important mainly near the absorption edge and at the top of the emission band.

At the heart of the study of topological insulators lies a fundamental\ndichotomy: topological invariants are defined in infinite systems, but surface\nstates as their main footprint only exist in finite systems. In the slab\ngeometry, namely infinite in two planar directions and finite in the\nperpendicular direction, the 2D topological invariant was shown to display\nthree different types of behaviour. The perpendicular Dirac velocity turns out\nto be a critical control parameter discerning between different qualitative\nsituations. When it is zero, the three types of behaviour extrapolate to the\nthree 3D topologically distinct phases: trivial, weak and strong topological\ninsulators. We show analytically that the boundaries between types of behaviour\nare topological phase transitions of particular significance since they allow\nto predict the 3D topological invariants from finite-thickness transitions.\nWhen the perpendicular Dirac velocity is not zero, we identify a new phase with\nsurface states but no band inversion at any finite thickness, disentangling\nthese two concepts which are closely linked in 3D. We also show that at zero\nperpendicular Dirac velocity the system is gapless in the 3D bulk and therefore\nnot a topological insulating state, even though the slab geometry extrapolates\nto the 3D topological phases. Finally, in a parameter regime with strong\ndispersion perpendicular to the surface of the slab, we encounter the unusual\ncase that the slab physics displays non-trivial phases with surface states but\nnevertheless extrapolates to a 3D trivial state.\n

Here we present two explicit counterexamples to the widely spread beliefs about an exclusive role of bimodality as the first-order phase transition signal. On the basis of an exactly solvable statistical model generalizing the statistical multifragmentation model of nuclei, we demonstrate that the bimodal distributions can naturally appear both in infinite and in finite systems without a phase transition. In the first counterexample a bimodal distribution appears in an infinite system at the supercritical temperatures due to the negative values of the surface tension coefficient. In the second counterexample we explicitly demonstrate that a bimodal fragment distribution appears in a finite volume analog of a gaseous phase. In contrast to the statistical multifragmentation model, the developed statistical model corresponds to the compressible nuclear liquid with the tricritical endpoint located at one third of the normal nuclear density. The suggested parameterization of the liquid phase equation of state is consistent with the L. Van Hove axioms of statistical mechanics and it does not lead to an appearance of the nonmonotonic isotherms in the macroscopic mixed phase region which are typical for the classical models of the Van der Waals type. Peculiarly, such a way to account for the nuclear liquid compressibility automatically leads to an appearance of an additional state that in many respects resembles the physical antinuclear matter.

The critical point in an infinite system is smeared out or “rounded” in a finite system into a “critical region.” This region has some structure associated with it, which is difficult to study except by means of simulations, i.e., Monte Carlo. I have begun a study of this structure in small two-dimensional Ising model systems, using the Markov property method.

In this paper we study the emission of a two-level atom in a radiation field in the case where one mode of the field is assumed to be excited initially, and where the system is assumed to be of infinite extent. (The restriction to a one-dimensional field, which has been made throughout this series, is not essential: It is made chiefly for ease of presentation of the mathematical methods.) An exact expression is obtained for the probability ρ (t) that the two-level quantum system is in the excited state at time t. This problem, previously unsolved in radiation theory, is tackled by reformulating the expression found in VII [J. Math. Phys. 16, 1013 (1975)] of this series for the time evolution of ρ (t) in a finite system in the presence of an extra photon, and then constructing the infinite-system limit. A quantitative assessment of the role of the extra photon and of the coupling constant in influencing the dynamics is obtained by studying numerically the expression derived for ρ (t) for a particular choice of initial condition. The study presented here casts light on the problem of time-reversal invariance and clarifies the sense in which exponential decay is universal; in particular, we find that: (1) It is the infinite-system limit which converts the time-reversible solutions of VII into the irreversible solution obtained here, and (2) it is the weak-coupling limit that imposes exponential form on the time dependence of the evolution of the system. The anticipated generalization of our methods to more complicated radiation-matter problems is discussed, and finally, several problems in radiation chemistry and physics, already accessible to exact analysis given the approach introduced here, are cited.

The statistical mechanics characterization of finite subsystems embedded in an infinite system is a fundamental question of quantum physics. Nevertheless, a full closed form for all required entropic measures does not exist in the general case, even for free systems, when the finite system in question is composed of several disjoint intervals. Here we develop a mathematical framework based on the Riemann–Hilbert approach to treat this problem in the one-dimensional case where the finite system is composed of two disjoint intervals and in the thermodynamic limit (both intervals and the space between them contains an infinite number of lattice sites and the result is given as a thermodynamic expansion). To demonstrate the usefulness of our method, we compute the change in the entanglement and negativity spectra namely the spectrum of eigenvalues of the reduced density matrix with or without time reversal of one of the intervals. We do this in the case that the distance between the intervals is much larger than their size. The method we use can be easily applied to compute any power in an expansion in the ratio of the distance between the intervals to their size. We expect these results to provide the necessary mathematical apparatus to address relevant questions in concrete physical scenarios, namely the structure and extent of quantum correlations in fermionic systems subject to local environments.